This document discusses fractions and decimals. It explains that all fractions can be written as decimals, and fractions with denominators that are powers of 2 or 5 will have terminating decimals, while other denominators will produce recurring decimals. It provides examples of converting terminating and recurring decimals to fractions. It also discusses changing fractions to decimals and vice versa. Finally, it defines rational and irrational numbers, providing examples of each, and asks students to identify numbers as rational or irrational, writing rational numbers as fractions in simplest form.

1. DEPTO MATEMÁTICAS 3ºESO FRACCIÓN GENERATRIZ
FRACTIONS AND DECIMALS
All fractions can be written as decimals .Example:
If the denominator of a fraction only has the prime factors 2 or 5,
its decimal terminates ( stops after a certain number of digits).
1
 1 : 5  0.2
5
Other denominators produce recurring decimals
Recurring decimals contain a group of repeating digits that “go for ever”.
They are shown by:
 A single dot above a single recurring digit
 A dot above the first and last digit of a set of recurring digits.
For example: 1 
 0.3333333333..........  0. 3
3
123  
 0.123123123.....  0.1 2 3
999
Changing terminating decimal as a fraction:
You can write a terminating decimal as a fraction.
Remember: a terminating decimal ends after a definite number of digits
Divide the number without the decimal point by the power of 10 that corresponds to
the number of digits in the decimal part.
Examples:
342 171
3.42  
100 50
325 65 13
0.0325   
1000 200 40
Changing recurring decimals to fractions
Recurring decimals can be converted into fractions
Let x= the recurring decimal.
Multiply both sides by the power of 10 that corresponds to the number of digits in
the recurring pattern.
E.g. by 101 =10 if only 1 digit recurs,
by 102  100 if 2 digits recur,
by 103  1000 if 3 digits recur,
IES OSTIPPO SECCIÓN BILINGÜE 1

2. DEPTO MATEMÁTICAS 3ºESO FRACCIÓN GENERATRIZ
 Subtract the original equation from the new equation.
 Solve the resulting equation for x.
 Make sure that the answer is a fraction in its simplest form.
Example:
Write down the fraction, in its simplest form, which is equal to these recurring
decimals.

a) 0. 4
x  0.444444444.......
Only 1 digit recurs, so, multiply both sides by 10
10 x  4.44444444444.......
Subtract the original equation from the new equation.
10 x  4.44444444444.......
 x  0.444444444444...
9x  4
4
x
9
 4
0.4 
9
b) x  0.575757.......
2 digits recur, so, multiply both sides by 100
100 x  57.575757.......
Subtract the original equation from the new equation.
100 x  57.575757.......
x  0.575757.......
99 x  57
Divide both sides by 99.
57 19
x  (in its simplest form)
99 33
  19
0.57 
33
IES OSTIPPO SECCIÓN BILINGÜE 2

3. DEPTO MATEMÁTICAS 3ºESO FRACCIÓN GENERATRIZ
REAL NUMBERS
Real numbers are either rational or irrational.
All terminating and recurring decimals are rational numbers.
An irrational number cannot be written as a fraction,
Irrational numbers include:
 Square roots of non-square numbers
 Cube roots of non-cube numbers.
Examples of irrational numbers are:
3
2 7  13
Examples:

1.- Show that 0.425 and 0. 2 are rational numbers.
0.425 is a terminating decimal.
425 17
0.425  
1000 40

0. 2 is a recurring decimal. x  0.2222222.....
10 x  2.2222...
 x  0.22222.......
9x  2
2
x
9
a
All terminating and recurring decimals can be written in the form so
b
they are all rational numbers.
2.- State whether each of the following are rational or irrational numbers.
Where a number is rational write it as a fraction in its simplest form.
3
0.6  7 36
6 3
a) 0.6   ……………………..rational
10 5
a
A rational number expressed in the form is in its simplest form if a and b
b
have no common factor.
IES OSTIPPO SECCIÓN BILINGÜE 3

4. DEPTO MATEMÁTICAS 3ºESO FRACCIÓN GENERATRIZ
b)   3.141592654... irrational
 is a non-recurring decimal and has no exact value.
3
c) 7  1.91293118..... irrational
3
7 is a non-recurring decimal and has no exact value.
d) 36  6 rational
IES OSTIPPO SECCIÓN BILINGÜE 4